Universal inversion formulas for recovering a function from spherical means

TL;DR

This paper develops universal back-projection formulas for reconstructing functions from spherical means over arbitrary convex domains, providing exact formulas for elliptical domains and approximate ones otherwise.

Contribution

It introduces universal inversion formulas applicable to any convex domain, with exact solutions for elliptical shapes, advancing imaging techniques involving spherical mean data.

Findings

Exact inversion formulas for elliptical domains in any dimension.

Universal formulas applicable to arbitrary convex domains.

The integral operator vanishes for elliptical domains, enabling precise reconstruction.

Abstract

The problem of reconstruction a function from spherical means is at the heart of several modern imaging modalities and other applications. In this paper we derive universal back-projection type reconstruction formulas for recovering a function in arbitrary dimension from averages over spheres centered on the boundary an arbitrarily shaped smooth convex domain. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed smoothing integral operator. For elliptical domains the integral operator is shown to vanish and hence we establish exact inversion formulas for recovering a function from spherical means centered on the boundary of elliptical domains in arbitrary dimension.